A025441 Number of partitions of n into 2 distinct nonzero squares.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0

Offset

0,66

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Michael Gilleland, Some Self-Similar Integer Sequences

Index entries for sequences related to sums of squares

Formula

a(A025302(n)) = 1. - Reinhard Zumkeller, Dec 20 2013

a(n) = Sum_{ m: m^2|n } A157228(n/m^2). - Andrey Zabolotskiy, May 07 2018

a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

a(n) = Sum_{i=1..floor((n-1)/2)} c(i) * c(n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020

a(n) = A000161(n) - A093709(n). - Andrey Zabolotskiy, Apr 12 2022

Mathematica

Table[Count[PowersRepresentations[n, 2, 2], pr_ /; Unequal @@ pr && FreeQ[pr, 0]], {n, 0, 107}] (* Jean-François Alcover, Mar 01 2019 *)

Prog

(Haskell)
a025441 n = sum $ map (a010052 . (n -)) $
                      takeWhile (< n `div` 2) $ tail a000290_list
-- Reinhard Zumkeller, Dec 20 2013
(PARI) a(n)=if(n>4, sum(k=1, sqrtint((n-1)\2), issquare(n-k^2)), 0) \\ Charles R Greathouse IV, Jun 10 2016
(PARI) a(n)=if(n<5, return(0)); my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)/2-issquare(n/2) \\ Charles R Greathouse IV, Jun 10 2016
(Python)
from math import prod
from sympy import factorint
def A025441(n):
    f = factorint(n).items()
    return -int(not (any((e-1 if p == 2 else e)&1 for p, e in f) or n&1)) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 0 # Chai Wah Wu, Sep 08 2022

Crossrefs

Cf. A060306 gives records; A052199 gives where records occur.

Cf. A000161, A000290, A010052, A025435, A157228, A053866, A145393, A093709.

Column k=2 of A341040.

Cf. A004439 (a(n)=0), A025302 (a(n)=1), A025303 (a(n)=2), A025304 (a(n)=3), A025305 (a(n)=4), A025306 (a(n)=5), A025307 (a(n)=6), A025308 (a(n)=7), A025309 (a(n)=8), A025310 (a(n)=9), A025311 (a(n)=10), A004431 (a(n)>0).

Sequence in context: A178602 A363711 A216279 * A286813 A176891 A219486

Adjacent sequences: A025438 A025439 A025440 * A025442 A025443 A025444

Keyword

nonn

Author

David W. Wilson

Status

approved